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15   WEAK TOPOLOGIES       79

4.   Let E be a separable real Frechet space and p a continuous seminorm on E.

(a)    Let (>„)„;> o be a total (12.13) free sequence in E, and let En be the subspace
of dimension n -f 1 generated by a0 , ai9 . . . , an . Show by induction on n that for
each n there exists a linear form/, on En such that/n is the restriction of fn+1 to £„,
fo(a0)=p(a0) and fn(x) ^p(x) for all #eEn. (Consider the hyperplane H in En_j
given by the equation fn-i(x) = fn-i(a0), and use Problem 3(d) in the plane which
is the quotient of En by the hyperplane H0 : fn-i(x) = 0 in Ert «i.)

(b)    Deduce from (a) that there exists a continuous linear form / on E such that
/(flo) = jp(flo) and \f(x)\ ^p(x) for all x e E (Hahn-Banach theorem for a separable
Frechet space). In particular, if E is a separable Banach space, then for each x e E
we have

(the norm on the dual E' being that defined in (5.7.3)). Deduce that if E and F are
separable Banach spaces, E' and F' their duals, and u : E -* F a continuous linear
mapping, then IMI = \\u\\.

(c)    Deduce from (b) the geometrical form of the Hahn-Banach theorem : given any
nonempty convex open set A in E, and a point x0 £ A, there exists a closed hyperplane
with equation g(x) — a = 0 such that g(x) > oc for all x e A, and g(xQ) <| a. (Reduce
to the case where 0 6 A and XQ is a frontier point of A. We can then assume that A is
symmetrical about 0 and therefore is the set of all x such that p(x) < 1, where p is
a continuous seminorm on E.)

(d)    Let A, B be two disjoint closed convex sets in E, one of which (say, A) is compact.
Show that there exists a closed hyperplane H, with equation g(x) — a = 0, which
strictly separates A and B: that is to say, such that g(x) > a for all A: e A and g(x) < oc
for all x e B.

(e)    Deduce from (d) that every closed convex set in E is an intersection of closed
half-spaces, and that every closed linear variety in E is an intersection of closed

(f)    Deduce that for a vector subspace F not to be dense in E it is necessary and
sufficient that there should exist a continuous linear form/ ^ 0 on E such that/Cx) = 0
for all x e F.

5. Let A be a convex set in a real vector space. A point a e A is said to be an extremal
point of A if there exists no line-segment containing more than one point, which is
contained in A and has a as an interior point: in other words, if the relations b e A,
c e A, a = Xb H- (1 — A)c, 0 < A < 1 imply b = c ~ a.

(a)    Let E be a Hilbert space, A a bounded closed convex set in E, 8 > 0 the diameter
of A, and a, b two points of A such that \\b — a\\ S> JA/15 • 8. Show that there exists
a supporting hyperplane H of A orthogonal to the vector b — a and such that the set
H n A has diameter gi$.

(b)    Deduce from (a) that A has at least one extremal point (proof by induction).

(c)    Show that A is the closed convex hull of the set of its extremal points (Krein-
Milman theorem for Hilbert space). (First deduce from (b) that every closed hyper-
plane of support of A contains at least one extremal point. Then argue by contradic-
tion, assuming that the closed convex hull B of the set of extremal points of A is
distinct from A; by using Problem 4(e), show that there would then exist a hyperplane
of support of A which did not meet B.) (Cf. Section 13.10, Problem 8.)onvex hull (Section 12.14, Problem 13) of the set of points ±en/n. Show